Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus

TL;DR

This paper demonstrates that the energy level fluctuations of high-genus hyperbolic surfaces follow GOE statistics almost surely, establishing a form of ergodicity in the spectral behavior across the moduli space.

Contribution

It proves that for most hyperbolic surfaces of high genus, the energy level fluctuations align with GOE predictions, extending previous ensemble results to almost sure convergence.

Findings

Energy variance matches GOE in the large genus limit

Typical surfaces exhibit GOE-like spectral fluctuations

Spectral ergodicity is established for high-genus hyperbolic surfaces

Abstract

We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. In this note we show that in this same limit, the energy variance for a typical surface is close to the GOE result, a feature called "ergodicity" in the random matrix theory literature.